Don't use FileHandler for testing... since FileHandler uses the filesystem... and so we will have a system test.

Use StreamHandler with a StringIO file.

Also, please try to avoid short variable names like "fh". 
In our test case, we only care to have a log handler over which we have control.

Here is an suggestion for creating the testing logger:

log_output = StringIO()
log_handler = StreamHandler(stream=log_output)

With this code, you don't need to mock the log_handler...

read the result using log_output.value() 

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Never forget about StringIO. StringIO one of your best friends in Python Testing Land :)